Duffing equation

The Duffing equation (or Duffing oscillator), named after Georg Duffing (1861–1944), is a non-linear second-order differential equation used to model certain damped and driven oscillators. The equation is given by $${\displaystyle {\ddot {x}}+\delta {\dot {x}}+\alpha x+\beta x^{3}=\gamma \cos(\omega t),}$$ where the (unknown) function ${\displaystyle x=x(t)}$ is the displacement at time t, ${\displaystyle {\dot {x}}}$ is the first derivative of ${\displaystyle x}$ with respect to time, i.e. velocity, and ${\displaystyle {\ddot {x}}}$ is the second time-derivative of ${\displaystyle x,}$ i.e. acceleration. The numbers ${\displaystyle \delta ,}$ ${\displaystyle \alpha ,}$ ${\displaystyle \beta ,}$ ${\displaystyle \gamma }$ and ${\displaystyle \omega }$ are given constants.

The equation describes the motion of a damped oscillator with a more complex potential than in simple harmonic motion (which corresponds to the case ${\displaystyle \beta =\delta =0}$); in physical terms, it models, for example, an elastic pendulum whose spring's stiffness does not exactly obey Hooke's law.

The Duffing equation is an example of a dynamical system that exhibits chaotic behavior. Moreover, the Duffing system presents in the frequency response the jump resonance phenomenon that is a sort of frequency hysteresis behaviour.